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mathematics
numerical analysis

Questions and Answers of Numerical Analysis

Determine the number of terms necessary to approximate cos x to 8 significant figure using the Maclaurin series approximation Calculate the approximation using a value of x = 0.3?. Write a program
Use 5-digit arithmetic with chopping to determine the roots the following equation with Eqs. (3.12) and (3.13) X2 – 5000.002 x + 10. Compute percent relative errors for your results.
How can the machine epsilon be employed to formulate a stopping criterion εs for your program? Provide an example.
The following infinite series can be used to approximate ex?: (a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion [(Eq. (4.7)] with xi?= 0 and h = x (b)
The Maclaurin series expansion for cos x is Starting with the simplest version, cos x = 1, add terms one at a time to estimate cos (?/3). After each new term is added, compute the-true and
Perform the same computation as in Prob. 4.2, but use the Maclaurin series expansion for the sin x to estimate sin (?/3).
Use zero- through third-order Taylor series expansions to predict ƒ (3) forf(x) = 25x3 – 6x2 + 7x – 88Using a base point at x = 1. Compute the true percent relative error εt for each
Use zero-through fourth-order Taylor series expansion to predict ƒ(2.5) for ƒ(x) = In x using a base point at x = 1. Compute the true percent relative error εt for each approximation. Discuss the
Use forward and backward difference approximations of O(h) and a centered differences approximation of O(h2) to estimate the first derivative of the function examined in Prob. 4.4.Evaluate the
Use a centered difference approximation of O(h2) to estimate the second derivative of the function examined in Prob.4.4. Perform the evaluation at x = 2 using step sizes of h = 0.25 and 0.125.
Recall that the velocity of the falling parachutist can be computed by [Eq. (1.10)], v(t) = 8m/c (1
Repeat Prob. 4.8 with g = 9.8, t = 6, c = 12.5 ± 1.5, and m = 50 ± 2.
The Stefan-Boltzmann law can be employed to estimate the rate of radiation of energy H from a surface, as in H = AeσT4. Where H is in watts, A = the surface area (m2), e = emissivity that
Repeat Prob. 4.10 but for a copper sphere with radius = 0.15 ± 0.01 m, e = 0.90 ± 0.05, and T = 550 ± 20.
Evaluate and interpret the condition numbersfor
Employing ideas from sec, 4.2 derive the relationships from Table 4.3.
Prove the Eq. (4.4) is exact for all values of x if ƒ(x) = ax2 + bx + c
Manning??s formula for a rectangular channel can be written as Where Q = flow (m3/s), n = a roughness coefficient, B = width (m), H = depth (m), and S = slope. You are applying this formula to a
If |x| < 1, it is known that 1/1 – x = 1 + x + x2 + x3 + . . .Repeat Prob. 4.2 for this series for x = 0.1.
A missile leaves the ground with an initial velocity ?0?forming an angle ?0?with the vertical as shown in figure The maximum desired altitude is ?R where R is the radius of the earth. The laws of
To calculate a planet’s space coordinate, we have to solve the function f(x) = x – 1 – 0.5 sin x Let the base point be α = xi = π/2 on the interval [0, π]. Determine the highest-order Taylor
Consider the function ƒ(x) = x3 – 2x + 4 on the interval [–2, 2] with h = 0.25. Use the forward, backward, and centered finite difference approximations for the first and second derivatives so
Determine the real roots of ƒ(x) = – 0.5x2 + 2.5x + 4.5:(a) Graphically.(b) Using the quadratic formula.(c) Using the iteration of the bisection method to determine the highest root. Employ
Determine the real root of ƒ(x) = 5x3 – 5x2 + 6x – 2:(a) Graphically(b) Using bisection to located the root. Employ initial guesses of xt = 0 and xu = 1 and iterate until the estimated error εa
Determine the real root of ƒ(x) = – 25 + 82x – 90x2 + 44x3 – 8x4 + 0.7x5:(a) Graphically.(b) Using bisection to determine the root to εs = 10%. Employ initial guesses of xt = 0.5 and xu =
(a) Determine the roots of ƒ (x) = – 12 – 21x + 18x2 -2.75x3 graphically. In addition, determine the first root of the function with(b) Bisection, and(c) False position. For (b) and (c) use
Locate the first nontrivial root of sin x = x3, where x is in radians. Use a graphical technique and bisection with the initial interval from 0.5 to 1. Perform the computation until εa is less than
Determine the positive real root of in (x4) = 0.7(a) Graphically,(b) Using three iterations of the bisection method, with initial guesses of xt = 0.5 and xu = 2, and(c) Using three iterations of the
Determine the real root of ƒ (x) = (0.8 – 0.3 x) / x:(a) Analytically.(b) Graphically.(c) Using there iterations of the false-position method and initial guesses of 1 and 3. Compute the
Find the positive square root of 18 using the false-position method to within εs = 0.5%. Employ initial guesses of xl = 4 and xu = 5.
Find the smallest positive root of the function (x is in radians) x2 |cos √x| = 5 using the false-position method. To locate the region in which the root lies, first plot this function for values
Find the positive real root of ƒ (x) = x4 – 8x3 – 35x2 + 450x – 1001 using the false-position method. Use initial guesses of xt = 4.5 and xa = 6 and performs five iterations. Compute both
Determine the real root of x3.5 = 80:(a) Analytically, and(b) With the false-position method to within εs = 2.5 %. Use initial guesses of 2.0 and 5.0
Given f(x) = – 2x6 – 1.5 x4 + 10 x + 2. Use bisection to determine the maximum of this function. Employ initial guesses of xt = 0 and perform iterations until the approximate relative error falls
The velocity υ of a falling parachutist is given by v = gm/c (1 – e – (c/m)t). Where g = 9.8 m/s2. For a parachutist with a drag coefficient c = 15 kg/s, compute the mass m so that the velocity
A beam is loaded as shown in figure. Use the bisection method to solve for the position inside the beam where there is nomoment.
Water is flowing in a trapezoidal channel at a rate of Q = 20 m3/s. The critical depth y for such channel must satisfy the equation 0 = 1 – Q2/gA3c B. Where g = 9.81 m/s2, Ac = the cross-sectional
You are designing a spherical tank Figure to hold water for a small village in a developing country. The volume of liquid it can hold can be computed as V = ?h2 [3R - h]/3 2. Where V = volume [m3], h
The saturation concentration of dissolved oxygen in fresh-water can be calculate with the equation Where Os??= the saturation concentration of dissolved oxygen in freshwater at l atm (mg/L) and
Integrate the algorithm outlined in Figure into a complete, user-friendly bisection subprogram. Among other things:(a) Place documentation statements throughout the subprogram to identify what each
Develop a subprogram for the bisection method that minimizes function evaluations based on the pseudocode from Figure Determine the number of function evaluations (n) per total iterations. Test the
Develop a user-friendly program for the false-position method. The structure of your program should be similar to the bisection algorithm outlined in Figure. Test the program by duplicating Example
Develop a subprogram for the false-position method that minimizes function evaluations in a fashion similar to Figure. Determine the number of function evaluations (n) per total iterations. Test the
Develop a user-friendly subprogram for the modified false-position method based on Figure. Test the program by determining the root of the function described in Example 5.6. Perform a number of run
Use simple fixed-point iteration to locate the roof ofƒ(x) = 2 sin(√x) – xUse an initial guess of x0 = 0.5 and iterate until εa ≤ 0.001%. Verify that the process is linearly convergent as
Determine the highest real root ofƒ (x) = 2x3 – 11.7x2 + 17.7x – 5(a) Graphically(b) Fixed-point iteration method (three iterations, x0 = 3). Note: Make certain that you develop a solution that
Use(a) Fixed-point iteration and(b) The Newton-rapshon method to determine a root of ƒ (x) = - x2 + 1.8x + 2.5 using x0 = 5. Perform the computation until εa is less than εs = 0.05%. Also perform
Determine the real roots of ƒ(x) = - 1 + 5.5x – 4x2 + 0.5x3:(a) Graphically and(b) Using the Newton-Raphson method to within εs = 0.01%.
Employ the Newton-Raphson method to determine a real root for ƒ(x) = - 1 + 5.5x – 4x2 + 0.5x3 using initial guesses of(a) 4.52 and(b) 4.54. Discuss and use graphical and analytical methods to
Determine the lowest real root of ƒ(x) = -12 – 21x + 18x2 – 2.4x3:(a) Graphically and(b) Using the secant method to a value of εs corresponding to three significant Figure.
Locate the first positive root ofƒ(x) = sin x + cos(1 + x2) – 1Where x is in radians, use four iterations of the secant method with initial guesses of(a) xi-1 = 1.0 and xi = 3.0;(b) xi-l = 1.5 and
Determine the real root of x3.5 = 80, with the modified secant method to within εs = 0.1% using an initial guess of x0 = 3.5 and δ = 0.01.
Determine the highest real root of ƒ(x) = 0.95x3 – 5.9x2 + 10.9x - 6:(a) Graphically.(b) Using the Newton-Raphson method (three iterations, xi = 3.5).(c) Using the secant method (three iterations,
Determine the lowest positive root of ƒ(x) = 8 sin(x) e-x – 1:(a) Graphically.(b) Using the Newton-Raphson method (three iterations, xi = 0.3).(c) Using the secant method (three iterations, xi-l =
The function x3 – 2x2 – 4x + 8 has a double root at x = 2. Use(a) The standard Newton-Raphson [Eq. (6.6)],(b) The modified Newton-Raphson [Eq. (6.9a)], and(c) The modified Newton-Raphson [Eq.
Determine the roots of the following simultaneous nonlinear equations using(a) Fixed-point iteration and(b) The Newton-Raphson method:y' = – x2 + x + 0.75y + 5xy = x2Employ initial guesses of x = y
Determine the roots of the simultaneous nonlinear equations (x – 4)2 + (y – 4)2 = 5, x2 + y2 = 16. Use a graphical approach to obtain your initial guesses. Determine refined estimates with the
Repeat Prob. 6.13 except for y = x2 + 1, y = 2 cos x
A mass balance for a pollutant in a well-mixed lake can be written as V dc/dt – W – Qc - kV√c. Given the parameter values V = 1 x 106m3, Q = 1 x 105 m3/yr, W = l x 106 g/yr, and k = 0.25
For Prob. 6.15, the root can be located with fixed-point iteration as Only one will converge for initial guesses of 2 < c < 6. Select the correct one and demonstrate why it will always work.
Develop a user-friendly program for the Newton-Raphson method based on Figure and sec. 6.2.3. Test it by duplicating the computation from Example 6.3.
Develop a user-friendly program for the secant method based on Figure and sec. 6.3.2. Test it by duplicating the computation from Example 6.6.
Develop a user-friendly program for the modified secant method based on Figure and sec. 6.3.2. Test it by duplicating the computation from Example 6.8.
Develop a user-friendly program for the two-equation Newton-Raphson method based on sec. 6.5. Test it by soling Example 6.10.
Use the program you developed in Prob. 6.20 to solve Probs.6.12 and 6.13 to within a tolerance of εs = 0.01%.
The “divide and average” methods, an old-time method for approximating the square root of any positive number α; can be formulated as x = x + a/x/2. Prove that this is equivalent to the
(a) Apply the Newton-Raphson method to the function ƒ(x) = tanh(x2 – 9) to evaluate its known real root at x = 3. Use an initial guess of x0 = 3.1 and take a minimum of four iteration.(b) Did the
The polynomial ƒ(x) = 0.0074x4 – 0.284x3 + 3.355x2 – 12.183x + 5 has a real root between 15 and 20. Apply the Newton Raphson method to this function using an initial guess of x0 = 16.15. Explain
Use the secant method on the circle function (x + 1)2 + (y – 2)2 = 16 to find a positive real root. Set your initial guess to xi = 3 and xi-1 = 0.5. Approach the solution from the first and fourth
You are designing a spherical tank (Figure) to hold water for a small village in a developing country. The volume of liquid it can hold can be computed as V = ?h2 [3R + h]/3. Where V = volume [ft3],
Divide a polynomial ƒ(x) = x4 – 7.5x3 + 14.5x2 + 3x – 20 by the monomial factor x – 2. Is x = 2 a root?
Divide a polynomial ƒ(x) = x5 – 5x4 + x3 - 6x2 – 7x + 10 by the monomial factor x –2.
Use Muller’s method to determine the positive real root of(a) ƒ(x) = x3 + x2 - 3x - 5(b) ƒ(x) = x3 – 0.5x2 + 4x – 3
Use Muller’s method or MATLAB to determine the real and complex roots of(a) ƒ(x) = x3 - x2 + 3x - 5(b) ƒ(x) = 2x4 + 6x2 –10(c) ƒ(x) = x4 - 2x3 + 6x2 – 8x + 8
Use Muller’s Bairstow’s method to determine the roots of(a) ƒ(x) = - 2 + 6.2x – 4x2 + 0.7x3(b) ƒ(x) = 9.34 - 21.97x + 16.3x2 - 3.704x3(c) ƒ(x) = x4 - 3x3 + 5x2 – x - 10
Develop a program to implement Muller’s method. Test it by duplicating Example 7.2.
Use the program developed in Prob. 7.6 to determine the real roots of Prob. 7.4a. Construct a graph (by hand or with Excel or some other graphics package) to develop suitable starting guesses.
Develop a program to implement Bairstoe’s method. Test it by duplicating Example 7.3.
Use the program developed in Prob. 7.8 to determine the roots of the equations in Prob. 7.5.
Determine the real roots of x3.5 = 80 with the Goal Seek capability of Excel or a library or package of your choice.
The velocity of a falling parachutist is given byv = 8m/c (1 – e–(e/m)t)Where g = 9.8 m/s2. For a parachutist with a drag coefficient c = 14 kg/s, compute the mass m so that the velocity is υ =
Determine the real roots of the simultaneous nonlinear equationsy = – x2 + x + 0.75y + 5xy = x2Employ initial guesses of x = y = 1.2 and use the Solver tool from Excel or a library or package of
Determine the real roots of the simultaneous nonlinear equations(x – 4)2 + (y – 4)2 = 5x2 + y2 = 16Use a graphical approach to obtain your initial guesses. Determine refined estimates with the
Perform the identical MATLAB operations as those in Example 7.7 or use a library or package of your choice to find all the roots of the polynomialƒ(x) = (x - 4)(x + 2)(x - 1)(x + 5)(x - 7)Note that
Use MATLAB or a library or package of your choice to determine the roots of the equations in Prob. 7.5.
Develop a subprogram to solve for the roots of a polynomial using the IMSL routine, ZREAL or a library or package of your choice. Test it by determining the roots of the equations from probs. 7.4 and
A two-dimensional circular cylinder is placed in a high-speed uniform flow. Vortices shed from the cylinder at a constant frequency, and pressure sensors on the rear surface of the cylinder detect
When trying to find the acidity of a solution of magnesium hydroxide in hydrochloric acid, we obtain the following equationA(x) = x3 + 3.5x2 – 40where x is the hydronium ion concentration. Find the
Consider the following system with three unknowns α, u, and υ:u2 – 2v2 = a2u + v = 2a2 – 2a – u = 0Solve for the real values of the unknown using:(a) The Excel Solver and(b) A symbolic
In control system analysis, transfer functions are developed that mathematically relate the dynamics of a system?s input to its output. A transfer function for a robotic positioning system is given
Develop an M-file function for bisection in a similar fashion to Figure. Test the function by duplicating the computations from Example 5.3 and 5.4.
Develop an M-file function for the false-position method. The structure of your function should be similar to the bisection algorithm outlined in Figure. Test the program by duplicating Example 5.5.
Develop an M-file function for the Newton-Raphson method based on Figure and Sec.6.2.3. Along with the initial guess, pass the function and its derivative as arguments. Test it by duplicating the
Develop an M-file function for the secant method based on Figure. And Sec.6.3.2. Along with the initial guess, pass the function as an argument. Test it by duplicating the computation from Example
Develop an M-file function for the modified secant method based on Figure. And Sec, 6, 3, 2. Along with the initial guess and the perturbation fraction, pass the function as an argument. Test it by
Perform the same computation as in Sec. 8.1, but for ethyl alcohol (a = 12.02 and b = 0.08407) at a temperature of 400 K and p of 2.5 atm. Compare your result with the ideal gas law. If possible, use
In chemical engineering, plug flow reactors (that is, those in witch fluid flows from one end to the other with minimal mixing along the longitudinal axis) are often used to convert reactants into
In a chemical engineering process, water vapor (H2O) is heated to sufficiently high temperature that a significant portion of the water dissociates, or splits apart, to form oxygen (O2) and hydrogen
The following equation pertains to the concentration of a chemical in a completely mixed reactor:If the initial concentration c0 = 5 and the inflow concentration cin = 12, compute the time required
A reversible chemical reaction 2A + B ⇌ C, can be characterized by the equilibrium relationshipK = cc/c2u cbWhere the nomenclature ci represents the concentration of constituent i. Suppose that we

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